Heat Transfer Analysis of Fins with Spine

Geometry Using Differential Transform Method

Melih Fidanoglu1, Guven Komurgoz

2, and Ibrahim Ozkol

1

1Faculty of Aeronautics and Astronautics, Istanbul Technical University, Istanbul, Turkey

2Faculty of Electrical and Electronical Engineering, Istanbul Technical University, Istanbul, Turkey

Email: {fidanoglum, komurgoz, ozkol}@itu.edu.tr

Abstract—In this study, a general expression for heat

distribution profile of a fin with spine geometry is found

using Differential Transform Method (DTM). By using this

general expression, fin efficiency, fin effectiveness, and

entropy generation are calculated for different profiles.

Based on efficiency, effectiveness and entropy generation

values, cylindrical fin profile is found to have the highest

heat transfer rate.

Index Terms—Differential Transform Method, DTM, spine

fin, heat distribution, fin efficiency, fin effectiveness,

entropy generation

I. INTRODUCTION

A wide array of engineering applications require

efficient energy transform mechanisms for rapid

movement of excessive heat. Extended surfaces are

widely used for transportation of heat. Fins are a type of

extended surface and they are used in various industries

such as in cooling turbines, heat exchangers and

electronic components.

There are a lot of extensive reviews about cooling by

fins are presented [1]-[7].

The entropy generation method was first introduced by

Bejan [8] as a measure of system performance and used

as a general criterion for judging the thermodynamic

performance of heat exchangers. He applied this method

to extended surfaces with four different specific

geometries, and used entropy generation minimization as

a performance criterion [9].

Differential Transform Method is a semi-analytical

numerical technique and it is used to solve differential

equations where finding an analytical solution may not be

possible. It is developed by Zhou in a study about

electrical circuits [10], it makes possible to obtain highly

accurate solutions for differential equations.

In the present study, DTM is used for calculating

temperature profile for spine fins with various profiles.

Fin efficiency and fin effectiveness are found to obtain

the optimum profile. The results are validated by making

an entropy generation minimization analysis.

Manuscript received July 11, 2015; revised October 21, 2015.

A. Equations of Heat Transfer for a Fin with Spine

Profile

The geometry of a general spine type fin is shown in

Fig. 1. Derivation of differential equation is carried out

by Kraus et al [2] and rewritten here for the explanation

of the problem.

Figure 1. The geometry of a general spine fin.

Profile for the fin is given as follows:

𝑓2(𝑥) =𝛿𝑏

2(

𝑥

𝑏)

(1−2𝑛)/(2−𝑛)

where 𝑛 < 0.5. (1)

The heat conduction into and out of the element 𝑑𝑥

through the cross section 𝑓1(𝑥) is:

𝑑𝑞

𝑑𝑥= 𝑘

𝑑

𝑑𝑥[𝑓1(𝑥)

𝑑𝜃

𝑑𝑥] (2)

The heat dissipation which occurs solely by convention

is:

𝑑𝑞

𝑑𝑥= ℎ𝑓3(𝑥)𝜃 (3)

Equaling right sides of these equations and rearranging

yields:

𝑓1(𝑥)𝑑𝜃

𝑑𝑥+

𝑑𝑓1(𝑥)

𝑑𝑥

𝑑𝜃

𝑑𝑥−

ℎ

𝑘𝑓3(𝑥)𝜃 = 0 (4)

By taking 𝑓1(𝑥) = 𝜋[𝑓2(𝑥)]2 and 𝑓3(𝑥) = 2𝜋𝑓2(𝑥) ,

the equation becomes:

[𝑓2(𝑥)]2 𝑑2𝜃

𝑑𝑥2 +𝑑

𝑑𝑥[𝑓2(𝑥)]2 𝑑𝜃

𝑑𝑥−

ℎ

𝑘𝑓2(𝑥)𝜃 = 0 (5)

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The boundary conditions are:

𝜃(𝑏) = 𝜃𝑏, 𝑑𝜃

𝑑𝑥|

𝑥=0= 0 (6)

II. THE SOLUTION METHOD AND DESIGN CRITERIA

A. Differential Transform Method (DTM)

Differential Transform Method is a technique based on

Taylor series expansion. A detailed analysis for the

theorem and various examples can be found in several

papers such as Arikoglu and Ozkol [11].

The transformation of a kth

derivative of a function at a

point 𝑥 = 𝑥0 can be found by the given formula:

𝐹(𝑘) =1

𝑘![

𝑑

𝑑𝑘𝑥𝑓(𝑥)]

𝑥=𝑥0

(7)

The inverse transformation can be found by the

following formula:

𝑓(𝑥) = ∑ 𝐹(𝑘)(𝑥 − 𝑥0)𝑘𝑁𝑘=0 for 𝑁 → ∞

(8)

The theorems that are used during the solution of fin

problem can be seen below:

Theorem1: If 𝑓(𝑥) = 𝑔(𝑥) ± ℎ(𝑥), then

𝐹(𝑘) = 𝐺(𝑘) ± 𝐻(𝑘).

(9)

Theorem 2: If 𝑓(𝑥) = 𝑐𝑔(𝑥), then

𝐹(𝑘) = 𝑐𝐺(𝑘).

(10)

Theorem3: If 𝑓(𝑥) =𝑑𝑛

𝑑𝑥𝑛 𝑔(𝑥), then

𝐹(𝑘) =(𝑛+𝑘)!

𝑘!𝐺(𝑛 + 𝑘).

(11)

Theorem4: If 𝑓(𝑥) = 𝑔(𝑥)ℎ(𝑥), then

𝐹(𝑘) = ∑ 𝐺(𝑘1)𝐻(𝑘 − 𝑘1)𝑘𝑘1=0 .

(12)

Theorem5: If 𝑓(𝑥) = 𝑔(𝑥 + 𝑎), then

𝐹(𝑘) = ∑ (ℎ1

𝑘) 𝑎ℎ1−𝑘𝐺(ℎ1)𝑁

ℎ1=𝑘

for 𝑁 → ∞.

(13)

B. Fin Efficiency and Fin Effectiveness

Fin Efficiency is the ratio of the actual fin heat transfer

rate to the heat transfer of the hypothetical fin if the entire

fin were at the base temperature. This ratio will be always

smaller than one. Its formula is given below:

𝜂𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 =𝑞𝑏𝑎𝑠𝑒

ℎ𝐴𝑓𝑖𝑛𝜃𝑏𝑎𝑠𝑒 (14)

Fin Effectiveness is the ratio of fin heat transfer rate to

the heat transfer of the base area as if there isn’t any fin.

This ratio will be always bigger than one. Its formula is

given below:

𝜂𝑒𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑒𝑛𝑒𝑠𝑠 =𝑞𝑏𝑎𝑠𝑒

ℎ𝐴𝑏𝑎𝑠𝑒𝜃𝑏𝑎𝑠𝑒 (15)

Total heat transfer of the fin through a cross section

𝐴(𝑥) = 𝜋[𝑓2(𝑥)]2 is defined as follows:

𝑞(𝑥) = 𝑘𝐴(𝑥)𝑑

𝑑𝑥𝜃(𝑥) = 𝑘𝜋[𝑓2(𝑥)]2 𝑑

𝑑𝑥𝜃(𝑥) (16)

C. Entropy Generation

The formula for entropy generation in a static fluid is

given as follows:

𝑆𝐺 =𝑞𝑏𝑎𝑠𝑒𝜃𝑏𝑎𝑠𝑒

𝑇∞2 (1+𝜃𝑏𝑎𝑠𝑒/𝑇∞)

(17)

where 𝑞𝑏𝑎𝑠𝑒 is the heat transfer rate at the base of the fin,

𝜃𝑏𝑎𝑠𝑒 is the temperature at the base of the fin, and 𝑇∞ is

the temperature of the medium.

III. RESULTS AND DISCUSSION

Eq. 5 is solved using the boundary conditions given in

Eq. 6 by DTM, and the results are compared to analytical

solutions for cylindrical, concave, conical, and convex

profiled spines. Taylor series for DTM are evaluated

around b/2. Variables used for the evaluation of Eq. 5 are

given as follows:

𝛿𝑏 = 0.01 (𝑚),

𝑘 = 32 (𝑊

𝑚∙ 𝐾),

𝑇𝑏𝑎𝑠𝑒 = 80 (°𝐶),

𝑏 = 0.1 (𝑚),

ℎ = 50 (𝑊

𝑚2∙ 𝐾)

𝑇𝑚𝑒𝑑𝑖𝑢𝑚 = 30

(18)

where 𝛿𝑏 is the base radius, 𝑏 are the base radius the fin

length, 𝑘 is the thermal conductivity, ℎ is the heat transfer

coefficient in, and 𝑇𝑏𝑎𝑠𝑒 and 𝑇𝑚𝑒𝑑𝑖𝑢𝑚 are the base and tip

temperatures.

Comparison of the temperature excess for various

profiles are shown at Table I and Table II. Here, �̅� = 𝑥/𝑏,

and temperature excess is defined as 𝜃 = 𝑇 − 𝑇𝑚𝑒𝑑𝑖𝑢𝑚.

TABLE I. COMPARISON OF DTM AND ANALYTICAL SOLUTION FOR

CYLINDRICAL AND CONVEX PARABOLIC SPINE

�̅� Cylindrical

Analytical

Cylindrical

DTM

Convex

Analytical

Convex

DTM

0.1 8.4096 8.4096 8.4901 8.7269

0.2 9.1941 9.1941 9.8498 9.9795

0.3 10.5562 10.5562 11.7735 11.8505

0.4 12.5816 12.5815 14.3056 14.3529

0.5 15.3973 15.3973 17.5383 17.5668

0.6 19.1805 19.1805 21.5996 21.6150

0.7 24.1686 24.1686 26.6524 26.6580

0.8 30.6752 30.6752 32.8980 32.8955

0.9 39.1091 39.1091 40.5818 40.5735

TABLE II. COMPARISON OF DTM AND ANALYTICAL SOLUTION FOR

CONICAL AND CONCAVE PARABOLIC SPINE

�̅� Conical

Analytical

Conical

DTM

Concave

Analytical

Concave

DTM

0.1 6.9178 7.2819 1.9208 2.0742

0.2 9.0898 9.2066 5.1238 5.1463

0.3 11.7154 11.7532 9.0959 9.0825

0.4 14.8655 14.8695 13.6677 13.6316

0.5 18.6201 18.6043 18.7443 18.6864

0.6 23.0688 23.0383 24.2632 24.1833

0.7 28.3120 28.2683 30.1793 30.0775

0.8 34.4622 34.4056 36.4582 36.3358

0.9 41.6449 41.5818 43.0726 42.9446

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As can be seen from the results, DTM solution is very

accurate at the points closer to the fin base, however this

is not the case for the points closer to the fin tip. It is

worth mentioning that the total heat transfer from the fin

is proportional to the heat gradient at the base. As a result,

from an engineering perspective, it can be said that this

method gives highly accurate results. Fig. 2 through Fig. 5 shows the temperature

distribution for DTM solutions for spine fins with

cylindrical profile, convex profile, conical profile, and

concave profile respectively.

Figure 2. Temperature distribution for a cylindrical profiled fin.

Figure 3. Temperature distribution for a convex profiled fin.

Figure 4. Temperature distribution for a conical profiled fin.

Figure 5. Temperature distribution for a concave profiled fin.

Comparisons of temperature distributions and total

heat flux distributions are given at Fig. 6 and Fig. 7

respectively. Total heat transfer can be measured by the

heat flux at the base. As can be seen from Fig. 6,

cylindrical fin has the highest total heat flux at the base,

thus it is the best design between those four types of fins

for heat removal. Also, total heat flux at the base

decreases as the n increases, and concave profiled fin is

the worst design considering only the total heat flux at the

base.

Figure 6. Temperature distribution for four different types of fins.

Figure 7. Heat flux distribution for four different types of fins.

Fin effectiveness and fin efficiency graphs can be seen

at Fig. 8 and Fig. 9 respectively. As can be seen from the

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figures, fin effectiveness increases with higher values of n,

and fin efficiency increases with lower values of n.

Having a higher value for both effectiveness and

efficiency for heat removal is desirable for a fin. Thus,

none of the designs satisfy both of these criteria at the

same time.

Figure 8. Fin effectiveness versus values of n.

Figure 9. Fin efficiency versus values of n.

In Fig. 10, Entropy Generation for various values of n

can be seen. For maximum heat dissipation from

surrounding medium, entropy generation value must be

maximized. As can be seen from the graph, entropy

generation increases as the value of n increases. Thus,

cylindrical profile has the highest entropy generation.

Figure 10. Entropy generation for various values of n.

IV. CONCLUSION

A wide range of applications such as air-cooled

automotive engines, air-conditioning systems, oil

industries, computer processors and other electronic

devices require extended surfaces for fast removal of

excessive heat.

In this study, a generalized solution for spine fins with

different geometries is sought. The DTM results for four

different types of fins are compared to analytical

solutions, and a good agreement is found at the base of

the fins. Fin efficiency, fin effectiveness and entropy

generation rates are found for various profiles. The results

show that, with increasing n, fin effectiveness and

entropy generation increases, and fin efficiency decreases.

Although cylindrical profile is found to be the best at heat

dissipation, further studies are required for profile

optimization, and fin array optimization with constraints

such as the amount of material used and the total base

area.

REFERENCES

[1] D. Q. Kern, Extended Surface Heat Transfer, McGraw-Hill, 1972. [2] F. Harahap and H. N. McManus, “Natural convection heat transfer

from horizontal rectangular fin arrays, ASME J. Heat Transfer,

vol. 89, 1967, pp. 32-38, 1967. [3] A. D. Kraus, A. Aziz, and J. Welty, Extended Surface Heat

Transfer, John Wiley & Sons, 2001.

[4] C. D. Jones and L. F. Smith, “Optimum arrangement of rectangular fins on horizontal surfaces for free convection heat

transfer,” ASME Journal of Heat Transfer, vol. 92, pp. 6-10, 1970. [5] A. Bar-Cohen, “Fin thickness for an optimized natural convection

array of rectangular fins,” ASME Journal of Heat Transfer, vol.

101, pp. 564-566, 1979. [6] D. W. Van De Pol and J. K. Tierney, “Free convective heat

transfer from vertical fin arrays,” Parts, Hybrids, and Packaging,

IEEE Transactions, vol. 10, no. 4, pp. 267-271, 1974. [7] R. C. Donvan and W. M. Roher, “Radiative and convective

conducting fins on a plane wall, including mutual irradiation,” ASME J. Heat Transfer, vol. 93, pp. 41-46, 1971.

[8] A. Bejan, “Second law analysis in heat transfer,” Energy, vol. 5,

no. 8-9, pp. 721-732, September 1980. [9] A. Bejan, “Fin geometry for minimum entropy generation in

forced convection,” Journal of Heat Transfer, vol. 104, no. 4, pp.

616-623, November 1982. [10] J. K. Zhou, Differential Transform and Its Applications in

Electrical Circuits, Huazhong Univ. Press, 1986.

[11] A. Arikoglu, “Solution of difference equations by using differential transform method,” Applied Mathematics and

Computation, vol. 174, no. 2, pp. 1216-1228, March 2006.

Melih Fidanoglu received his B.Sc. degree in

Mechatronics Engineering from Sabanci

University at 2005, and received his M.Sc. in

Defense Technologies from Istanbul

Technical University at 2011. He is currently

working toward the PhD degree in Aeronautical Engineering at the Faculty of

Aeronautics and Astronautics of Istanbul

Technical University. His research interests are numerical methods, heat transfer, and fluid

dynamics.

Guven Komurgoz completed her BS (1991),

MS (1995), and PhD (2001) all in electrical engineering at the Istanbul Technical

University, Turkey. She is currently an

associate professor of electrical engineering. She is interested in Heat Transfer, Numerical

Methods, Design of Transformer and

Electrical Machines.

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International Journal of Mechanical Engineering and Robotics Research Vol. 5, No. 1, January 2016

© 2016 Int. J. Mech. Eng. Rob. Res.

Ibrahim Ozkol received his B.Sc., M.Sc., and PhD degrees in Aeronautical Engineering

from Istanbul Technical University (ITU),

Department of Aeronautical Engineering. He is currently a professor of Aeronautics in the

Department of Aeronautical Engineering at

Istanbul Technical University. His research interests include dynamics and kinematics of

parallel mechanisms, rotating flow and

thermal fields, and engineering problems in applied mathematics.

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